Graph each function over a two-period interval.
y = 1 + ...

Question

Graph each function over a two-period interval.

y = 1 + tan x

Accepted Answer

Hey, everyone in this problem, we're asked to graph the following function and to consider two periods, the function we're given is Y is equal to four plus tangent of X. We're given this blank graph here where we can draw this function out. Now we're O gra two periods. So let's start by figuring out what the period of this function is. OK. So our function Y is equal to four plus tangent of X. Now the period is gonna be related to the B value. Recall that the B value is that coefficient associated or on the X inside of the brackets inside of our trigonometric function. Now, in this case, we just have tangent of X. So our B value is just going to be equal to one if we have a B value of one, it means that the period of this function we have is the same as the regular period of tangent of X. OK. So our period, it's always going to be that regular period of tangent of X which is pi divided by our B value which in this case, just pie. OK. So we have the exact same period now that makes sense. OK. We have tangent of X and then we're just adding four. OK. So we have our regular tangent function and then we're shifting it vertically by four. Mhm. All right. So we have our period of pie, we want graph for two periods. Now we can graph any two periods we like. But if we take a look at the graph, we were given, we already have some labels on the X axis. On the far left, we have negative pi divided by two. On the far right, we have three pi divided by two. And so we're gonna graph on that interval, that's gonna be two pi K or those two periods we're interested in. So we're gonna graph from negative pi divided by two, two, the RP I might have my two. All right. Now we're dealing with a tangent function. So we know that there's going to be asymptotes. So let's look at those vertical asymptotes figure out where they are. And that's gonna help us draw this di this diagram up. So we are going to find our vertical asymptotes or VA S. Now we have the function Y is equal to four plus tangent of X. Now, when we have tangent functions or cotangent functions, and we're looking for asymptotes, it can be easier to split it out into sine and cosine. OK? Because we know that tangent is sine divided by cosine. And when we're looking for vertical asymptotes, we're interested in what's happening in the denominator of our function. So let's do that. Now, we can write this as four what sign of X divided by cosine of X? And we want to write this as one entire fraction, we can do that by using a common denominator. So on our four term, we can multiply the numerator and denominator by cosine and then write it as four cosine of X. What sign of X all divided by cosine X? When we do this, we can see more clearly where those vertical asymptotes come from. If the denominator is equal to zero, our function will not exist there. We can't divide by zero and that's where those vertical asymptotes come. OK. So in this case, now we have our denominator and so we can clearly see that the vertical asymptotes are going to occur when cosine of X is equal to zero. When does cosine of X equal zero? OK. Well, you can imagine looking at your regular cosine curve or thinking about the values associated with cosine. And we know that cosine of X is zero if X is pi divided by two, some integer multiple. OK, some odd multiple. So we take pi divided by two and we add KP OK, isn't it? OK. So we're thinking about pi divided by 23, pi divided by 25 pi divided by two, et cetera, et cetera. So these are in general where vertical asymptotes occur. Let's think about the specific vertical asymptotes we have within the interval or graphic. Now, we know that we have some negative X values that we're graphing. So let's think about how we would get that for X value of our vertical Asymptote and we'll need K to be negative. So let's start with K is equal to negative one. If we substitute K equals negative one, we get that X is going to be equal to pi divided by two minus pi which is going to be equal to negative pi divided by two. So that's the first Asymptote we have that we are gonna grow and that's that far left end of the plane that we were given. OK. Let's move to the next K value. If K is equal to zero, what do we get? Or we just get that X is equal to pi divided by two. Yeah, that's within the interval that we're graphic. Let's move to the next K value K is equal to one. OK. If K is equal to one, we get that X is going to be pi divided by two plus pi which is three pi divided by two. And this is that far right of our interval. OK. So we don't need to look for any more K values. Anything else greater will be outside of the interval or graphic. So we have three asymptotes within the interval. We're graphing, we're gonna go ahead and add those to our graph right now. We're gonna draw them as dashed vertical lines. So at negative pi divided by two, we have a vertical Asymptote. At pi divided by two, we have a vertical Asymptote. And at three pi divided by two, we have a vertical Asymptote. And what you can see is that these vertical asymptotes kind of break those periods up. And as it breaks our interval up into those two periods, now we mentioned at the beginning that this function is the regular tangent curve that's been shifted upwards for units. OK. So we can think about that to graph it. And another way we can do this is by doing a table of value. OK. So we use a table of values, we're gonna be able to see exactly where those points have shifted to and we're gonna be able to draw it. So our table of values we're gonna do for the first interval, OK? Or the first period, sorry. OK. So the first period, we're gonna look at a table of values, we could do the same for the second period. But we know that when we're dealing with a periodic function like this, the second period is just a repeat of the first. OK? And it repeats and repeats and repeats. So we only need to do the first period and then we'll be able to copy it for the second. So let's do a table of values. We have our X values that we're interested in. And our Y values that we're interested in. Now, the X values we're gonna start at the far left negative pi divided by two and we have negative pi divided by four zero, pi divided by four and pi divided by two. Now pi uh negative pi divided by two and pi divided by two. This is where our vertical asymptotes occur. OK. So those are going off to either positive or negative infinity at that point, our function is going off to positive or negative infinity. And so we're not going to calculate the values there, but we will do these other points and the other points are gonna help us figure out whether we're going to positive infinity or negative infinity at each atom. OK. So if X is negative pi divided by four, we get that our function is four plus tangent of negative pi divided by four. Right now, tangent of negative pi divided by four is negative one. So we get four minus one which gives us a value of three. If X is zero, we get four plus tangent of zero, tangent of zero is just zero. And so we end up with four. And if X is pi divided by four, we get four plus tangent of pi divided by four tangent of P of pi divided by four is one. So we get four plus one and this gives us pi, OK. So these are all values that should know a tangent of negative pi divided by 40 pi divided by four. These are those standard um angles that we have these trigonometric functions. All right. So we have these three values, let's add them to our graph. So at negative pi divided by four, our function has a value of three at zero, our function has a value of four and that pi divided by four, it has a value of five and we can connect these like a tangent curve. We know that it follows that same typical tangent behavior. It's gonna look something like this. OK. So as we approach our vertical Asymptote at negative pi divided by two from the right side, our functions approaching negative infinity as we approach pi divided by two from the left side, we're approaching positive infinity. So this is our first period and we can repeat the same in the second period. So halfway through that second period, we're gonna be at four at three pi divided by four, we're gonna be at three, five pi divided by four at five. And again, connecting these like a tangent curve. And we can see that this function has just repeated for that second period. All right. So this is our graph of Y is equal to four plus tangent of X. Thanks everyone for watching. I hope this video helped you in the next one.

Graph each function over a two-period interval.
y = 1 + tan x

Key Concepts

Periodicity of the Tangent Function

Vertical Asymptotes of Tangent

Vertical Translation of Functions

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